Gravitation
Chapter Outline
ENGINEERING PHYSICS I
Newton’s Law of Universal Gravitation
PHY 303K
The Measurement of G
Chapter 9: Gravitation
Circular Orbits
Elliptical Orbits; Kepler’s Laws
Maxim Tsoi
Energy in Orbital Motion
Physics Department,
The University of Texas at Austin
http://www.ph.utexas.edu/~tsoi/303K.htm
303K: Ch.9
303K: Ch.9
Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation
Unification of “earthly” and “heavenly” motions
Unification of “earthly” and “heavenly” motions
Every particle in the Universe attracts every other particle with a
force that is directly proportional to the product of their masses and
inversely proportional to the square of the distance between them
Every particle in the Universe attracts every other particle with a
force that is directly proportional to the product of their masses and
inversely proportional to the square of the distance between them
Fg  G
m1m 2
r2
Fg  G
• G is the universal gravitational constant = 6.67310-11 Nm2/kg2

mm
F12  G 1 2 2 r̂12
r


F21   F12
m1m 2
r2
• Gravitational force is a field force  always exist between two particles
regardless of the medium that separates them
• Fg~1/r2  decreases rapidly with increasing separation
• a finite-size spherically symmetric mass distribution produces the same
gravitational force as if the entire mass were concentrated at its center
Fg  G
303K: Ch.9
303K: Ch.9
Newton’s Law of Universal Gravitation
Measuring the Gravitational Constant
Unification of “earthly” and “heavenly” motions
Experiment by Henry Cavendish in 1798
• Two spheres (each of mass m) fixed to the ends of a light horizontal
Every particle in the Universe attracts every other particle with a
force that is directly proportional to the product of their masses and
inversely proportional to the square of the distance between them
Fg  G
Fg  G
MEm
 ma
r2

a G
2
rod suspended by a thin metal wire
• Two large spheres (each of mass M) are placed near the small ones
m1m 2
r2
ME
r2
2
a M 1 rM2  RE   6.37  106 m 
  2.75  104
 


g 1 RE2  rM   3.84  108 m 
a M  2.75  10 4 9.8m / s 2  2.70  103 m / s 2




• The attractive force between smaller and larger spheres causes the
rod to rotate and twist the wire

v 2 2rM / T 
4 2 3.84  108 m
aM 


 2.72  10 3 m / s 2
2
rM
rM
2.36  106 s
2

303K: Ch.9
MEm
RE2

• The angle of rotation is measured by deflection of light beam for
different masses at various separations
303K: Ch.9
1
Free-Fall Acceleration
Kepler’s Laws and the Motion of Planets
g and the Gravitational Force
Kepler’s analysis of planetary motion is summarized in three statements
• The magnitude of the force acting on a freely falling object of
mass m near the Earth’s surface:
mg  G
MEm
RE2

g G
(1)
All planets move in elliptical orbits with the Sun at one focus
ME
RE2
• For an object located a distance h above the Earth’s surface:
M m
MEm
Fg  G E2  G
2
r
RE  h



g G
R
ME
E
h

(2)
The radius vector drawn from the Sun to a planet sweeps out
equal areas in equal time intervals
(3)
The square of the orbital period of any planet is proportional to
the cube of the semimajor axis of the elliptical orbit
2
• g decreases with increasing altitude!
303K: Ch.9
303K: Ch.9
Kepler’s Laws and the Motion of Planets
Kepler’s Laws and the Motion of Planets
First Law
Second Law
All planets move in elliptical orbits with the Sun at one focus
The radius vector drawn from the Sun to a planet sweeps out equal
areas in equal time intervals
ec a
• Eccentricity of an ellipse 
• consequence of angular momentum conservation
Earth: 0.017; Pluto: 0.25; Comet Halley: 0.97
  
  r F  0
  
 
L  r  p  M P r  v  const
• Torque of a central force:
• Angular momentum:
 
 
dA  12 r  dr  12 r  v dt 
• Aphelion  the point where the planet is farthest away
L
dt
2M P
dA
L

 const
dt 2 M P
from the Sun (apogee for an object orbiting the Earth)
• Perihelion  the point nearest the Sun (perigee for an
object orbiting the Earth)
303K: Ch.9
303K: Ch.9
Kepler’s Laws and the Motion of Planets
The Gravitational Field
Third Law
How objects interact when they are not in contact?
The square of the orbital period of any planet is proportional to the
cube of the semimajor axis of the elliptical orbit
v
• The orbital speed:
 4 2
T 2  
 GM S
303K: Ch.9
 3
a  K S a 3

• When a particle of mass m is placed at a point where the
GM S M P M P v 2

r2
r
• Newton’s 2nd law:
• The period:
• Gravitational field exists at every point in space
2r
T
 4 2
T 2  
 GM S
KS 
gravitational field is g, the particle experience
 a force Fg=mg
• Gravitational field is defined as
 3
r  K S r 3

 Fg
g
m
The Earth’s gravitational field:

 Fg
GM
g
  2 E rˆ
m
r
4 2
 2.97  1019 s 2 / m 3
GM S
303K: Ch.9
2
Gravitational Potential Energy
Gravitational Potential Energy
Any central force is conservative!
Work done by the gravitational force
• The change in the gravitational potential
• Central force  directed along a radial line
to a fixed center and has a magnitude that
energy of a particle associated with a
depends only on the radial coordinate r
given particle’s displacement is defined as
the Earth-particle system
r≥RE
the negative of the work done by the
• Conservative force  work it does on an
gravitational force on the particle during
object moving between two points is
the displacement
independent of the path
U  U f  U i    F r dr
rf
 
dW  F  dr  F r dr
ri
U f  U i  GM E m 
W   F r dr
rf
rf
ri
The path is broken into a series
of radial segments and arcs
ri
U r   
• The work depends only on the initial and
final values of r  force is conservative
303K: Ch.9
1 1
dr
 GM E m 
r
r2
 f ri




F r   
GM E m
r
GM E m
r2
303K: Ch.9
Gravitational Potential Energy
Energy Considerations
Binding energy
in planetary and satellite motion
• An object of mass m is moving around a massive object
• The gravitational potential energy associated
of mass M (at rest)
with any pair of particles of masses m1 and m2
separated by a distance r is
U r   
• The total mechanical energy of the two-object system:
Gm1m2
r
E  K U 
• For three or more particles:
m m m m m m 
U total  U12  U13  U 23  G 1 2  1 3  2 3 
r13
r23 
 r12
• Circular orbit:
mv 2
GMm
 ma 
r2
r
• Elliptical orbit:
E
• The absolute value of Utotal represents the work
needed to separate the particles by an infinite
distance  binding energy of the system
Example 9.6
303K: Ch.9
GMm
2a

mv 2 GMm

2
r
mv 2 GMm

2
2r

E 
GMm
2r
The total mechanical energy of
a bound system is negative!
Both the total energy and the total angular momentum of a
gravitationally bound, two-object system are constants of the motion
303K: Ch.9
Energy Considerations
Energy Considerations
Escape speed
Black holes
• An object of mass m is projected vertically upward from the Earth’s surface
are remains of stars that have collapsed under their own gravitational force
• Escape speed  the minimum speed the object must have in order to
approach an infinite separation distance from the Earth
• Very massive star  supernova 
remaining central core continues to
mv i2 GM E m
GM E m


2
RE
rmax
collapse  depending on its mass:
<1.4 MS  white dwarf star
 1
1 

v i2  2GM E 

 RE rmax 
vesc 
>1.4 MS  neutron star R~10 km
>~3 MS  black hole
• Escape speed exceeds the speed of
2GM E
RE
light  the object appears to be
black
Independent of the mass
of the object!
303K: Ch.9
Example 9.8
Any event occurring within the
boundary of the Schwarzschild
radius is invisible to an outside
observer
303K: Ch.9
3
SUMMARY
Gravitation
• Newton’s law of universal gravitation:
Fg  G
• Kepler’s laws of planetary motion:
m1m2
r2
(1) All planets move in elliptical orbits with the Sun at one focus
(2) The radius vector drawn from the Sun to a planet sweeps
out equal areas in equal time intervals
(3) The square of the orbital period of any planet is proportional
to the cube of the semimajor axis of the elliptical orbit
• The gravitational field:
 Fg
g
m
• The gravitational potential energy: U r   
• The total energy of the bound system:
• Escape speed:
vesc
2GM E

RE
Gm1m 2
r
E
GMm
2a
303K: Ch.9
4